Spectral zeta functions of fractals and the complex dynamics of polynomials

Teplyaev, Alexander

doi:10.1090/S0002-9947-07-04150-5

Spectral zeta functions of fractals and the complex dynamics of polynomials
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Trans. Amer. Math. Soc. 359 (2007), 4339-4358 Request permission

Abstract:

We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpiński gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpiński gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

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